Disk drive determining frequency response of actuator near servo sample frequency

ABSTRACT

A disk drive is disclosed comprising a servo control system configured to control an actuator for actuating a head over a disk. Servo sectors are sampled at a servo sample frequency f s  to generate a position error signal PES(k) that is filtered with a compensator to generate a first control signal u 1 (k). A first discrete-time sinusoid comprising a sinusoid frequency of f s +Δf is added to the first control signal u 1 (k) to generate a second control signal u 2 (k). The second control signal u 2 (k) is applied to the actuator, and a frequency response of the actuator is determined at the frequency |Δf|. The frequency response of the actuator is determined at the frequency f s +Δf based at least in part on the frequency response of the actuator at the frequency |Δf| and a measured signal of the servo control system when applying the second control signal u 2 (k) to the actuator.

BACKGROUND

Disk drives comprise a disk and a head connected to a distal end of an actuator arm which is rotated about a pivot by a voice coil motor (VCM) to position the head radially over the disk. The disk comprises a plurality of radially spaced, concentric tracks for recording user data sectors and servo sectors. The servo sectors comprise head positioning information (e.g., a track address) which is read by the head and processed by a servo control system to control the actuator arm as it seeks from track to track.

FIG. 1 shows a prior art disk format 2 as comprising a number of servo tracks 4 defined by servo sectors 6 ₀-6 _(N) recorded around the circumference of each servo track. Each servo sector 6 _(i) comprises a preamble 8 for storing a periodic pattern, which allows proper gain adjustment and timing synchronization of the read signal, and a sync mark 10 for storing a special pattern used to symbol synchronize to a servo data field 12. The servo data field 12 stores coarse head positioning information, such as a servo track address, used to position the head over a target data track during a seek operation. Each servo sector 6 _(i) further comprises groups of servo bursts 14 (e.g., N and Q servo bursts), which are recorded with a predetermined phase relative to one another and relative to the servo track centerlines. The phase based servo bursts 14 provide fine head position information used for centerline tracking while accessing a data track during write/read operations. A position error signal (PES) is generated by reading the servo bursts 14, wherein the PES represents a measured position of the head relative to a centerline of a target servo track. A servo controller processes the PES to generate a control signal applied to a head actuator (e.g., a voice coil motor) in order to actuate the head radially over the disk in a direction that reduces the PES.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a prior art disk format comprising a plurality of servo tracks defined by servo sectors.

FIG. 2A shows a disk drive according to an embodiment comprising a head actuated over a disk by an actuator comprising a frequency response.

FIG. 2B shows a servo control system according to an embodiment wherein a discrete-time sinusoid comprising a sinusoid frequency of f_(s)+Δf is added to a first control signal u1(k) to generate a second control signal u2(k).

FIG. 2C is a flow diagram according to an embodiment for determining a frequency response of the actuator at the frequency f_(s)+Δf based at least in part on the frequency response of the actuator at the frequency |Δf| and a measured signal of the servo control system when applying the second control signal u2(k) to the actuator.

FIG. 3A represents a closed-loop sampled servo control system according to an embodiment.

FIG. 3B illustrates an embodiment for deriving the effective transfer function of the closed-loop sampled servo control system.

FIG. 4A shows an equivalent system to FIG. 2B when evaluated relative to a position error signal PES(k) of the servo control system.

FIG. 4B shows an embodiment for determining a frequency response of the actuator at the frequency f_(s)+Δf based on the PES(k) of the servo control system.

FIG. 4C shows an embodiment for determining a frequency response of the actuator at the frequency f_(s)+Δf based on the first control signal u1(k).

FIG. 4D shows an alternative embodiment for determining a frequency response of the actuator at the frequency f_(s)+Δf based on the first control signal u1(k).

DETAILED DESCRIPTION

FIG. 2A shows a disk drive according to an embodiment comprising a head 16, a disk 18 comprising a plurality of servo tracks 20, wherein each servo track comprises a plurality of servo sectors 22 ₀-22 _(N). The disk drive further comprises an actuator (e.g., voice coil motor (VCM) 24 and/or a microactuator 26) configured to actuate the head 16 over the disk 18, wherein the actuator comprises a frequency response. The disk drive further comprises control circuitry 28 comprising a servo control system (FIG. 2B) configured to control the actuator. The control circuitry 28 is configured to execute the flow diagram of FIG. 2C, wherein the servo sectors are sampled at a servo sample frequency f_(s) to generate a position error signal PES(k) 30 (block 32). The PES(k) 30 is filtered with a compensator C(z) 34 to generate a first control signal u1(k) 36 (block 38). A first discrete-time sinusoid 40 comprising a sinusoid frequency of f_(s)+Δf is added to the first control signal u1(k) 36 to generate a second control signal u2(k) 42 (block 44), wherein the second control signal u2(k) 42 is applied to the actuator P(s) 46 (block 48). The frequency response of the actuator P(s) 46 is determined at the frequency |Δf| (block 50). The frequency response of the actuator P(s) 46 is determined at the frequency f_(s)+Δf based at least in part on the frequency response of the actuator P(s) 46 at the frequency |Δf| and a measured signal of the servo control system when applying the second control signal u2(k) 42 to the actuator P(s) 46 (block 52).

In the embodiment of FIG. 2A, the control circuitry 28 processes a read signal 54 emanating from the head 16 to demodulate the servo sectors 22 ₀-22 _(N) to generate an actual position of the head that is subtracted from a reference position to generate the PES(k) 30. The control circuitry 28 generates a control signal 56 applied to the VCM 24 which rotates an actuator arm 58 about a pivot to actuate the head 16 radially over the disk 18 in coarse movements. In one embodiment, the control circuitry 28 may also generate a control signal 60 applied to a microactuator 26 to actuate the head 16 over the disk 18 in fine movements. The servo sectors 22 ₀-22 _(N) may comprise any suitable head position information, such as a track address for coarse positioning and servo bursts for fine positioning. The servo bursts may comprise any suitable pattern, such as an amplitude based servo pattern or a phase based servo pattern (FIG. 1).

In one embodiment, it may be desirable to measure a frequency response of the actuator P(s) 46 for actuating the head 16 over the disk 18 in order, for example, to identify resonant frequencies of the actuator P(s) 46. In one embodiment, the servo control system may be modified based on the identified resonant frequencies, such as by adding and/or modifying notch filters that attenuate the frequency response at the resonant frequencies. In another embodiment, the identified resonant frequencies may be used to identify defective servo components, such as a defective VCM 24 or microactuator 26, so that the disk drive may be discarded or reworked to replace the defective components.

Any suitable technique may be employed to measure the frequency response of the actuator P(s) 46. In one embodiment, the control circuitry 28 executes a signal processing algorithm capable of measuring the frequency response of the servo control system at frequencies higher than half the second servo sample frequency. Such a signal processing algorithm may include an anti-aliasing multi-rate (Nx) bode algorithm which is understood with reference to FIGS. 3A and 3B. FIG. 3A represents a closed-loop sampled servo control system where G_(p)(jω) represents the plant under test (e.g., a compensator C(z) and actuator P(s)), r(t) represents a reference input, y(t) represents the sampled output (e.g., the PES measured at each servo sector), and Gh(jω) represents a zero order hold function. In one embodiment, the frequency response of the closed-loop servo control system shown in FIG. 3A is measured at discrete frequencies (e.g., frequency ω₀) by injecting a sinusoid having a frequency ω₀ as the reference input R(jω). The effective transfer function Heff(jω) may be derived as shown in FIG. 3B where T represents the servo sample period. The term H_(Σ)(jω) represents the discrete-time transfer function of the closed-loop system evaluated at z=e^(jωT), and ω_(s) represents the servo sample frequency. Since the effective transfer function Heff(jω) does not exhibit aliasing (anti-aliasing) it may be measured at any frequencies, including frequencies beyond half the servo sample frequency (the Nyquist frequency). However, when using the above-described multi-rate (Nx) bode algorithm, the frequency response is undefined when the frequency of the reference input R(jω) is proximate an integer multiple of the servo sample frequency (kω_(s)).

Accordingly, in one embodiment the frequency response of the actuator P(s) 46 may be measured proximate the servo sample frequency by injecting a discrete-time sinusoid 40 comprising a sinusoid frequency of f_(s)+Δf into the servo control system as shown in FIG. 2B. FIG. 4A shows an equivalent system to FIG. 2B when evaluated relative to the PES(k) 30. Due to the sampler in FIG. 4A, injecting the discrete-time sinusoid 40 as shown in FIG. 2B results in an alias signal d_(alias) 62 being fed back to the compensator C(z) 34 of the form: d _(alias) =A ₀ |P(j2π(f _(s) +Δf))|Sin(2kπΔfT _(s)+α)  (1) where α represents the phase response of P(j2π(f_(s)+Δf)). When injecting the discrete-time sinusoid 40 as shown in FIG. 2B, the alias signal d_(alias) 62 may be calculated by measuring the PES(k) 30: d _(alias)=(1+P(j2π|Δf|)C(j2π|Δf|))PES  (2) or by measuring the first control signal u1(k) 36:

$\begin{matrix} {{\left. {d_{alias} = {X\left( {j\; 2\;\pi{{\Delta\; f}}} \right)}} \right)U\; 1}{{X(s)} = {\left( {1 + {{P(s)}{C(s)}}} \right)/{C(s)}}}} & (3) \end{matrix}$

By equating the above equation (1) and equation (2), the frequency response of the actuator P(s) 46 at the frequency f_(s)+Δf may be determined according to the equation shown in FIG. 4B:

${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{{PES}\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}\left( {1 + {{P\left( {{\Delta\; f}} \right)}{C\left( {{\Delta\; f}} \right)}}} \right)}$ where P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf, PES(|Δf|) represents the frequency domain representation of the PES(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator, S₂(|Δf|) represents a frequency domain representation of a second discrete-time sinusoid at a sinusoid frequency of |Δf|, P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|, and C(|Δf|) represents a frequency response of the servo compensator at the frequency |Δf|.

By equating the above equation (1) and equation (3), the frequency response of the actuator P(s) 46 at the frequency f_(s)+Δf may be determined according to the equation shown in FIG. 4C:

${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{U\; 1\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}\left\lbrack \frac{1 + {{P\left( {{\Delta\; f}} \right)}{C\left( {{\Delta\; f}} \right)}}}{C\left( {{\Delta\; f}} \right)} \right\rbrack}$ where P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf, U1(|Δf|) represents the frequency domain representation of the first control signal u1(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator, S₂(|Δf|) represents a frequency domain representation of a second discrete-time sinusoid at a sinusoid frequency of |Δf|, P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|, and C(|Δf|) represents a frequency response of the servo compensator at the frequency |Δf|. When |Δf| is a low frequency, |P(s)(C(s)|>>1 such that X(s)≈P(s) and therefore the frequency response of the actuator P(s) 46 at the frequency f_(s)+Δf may be determined according to the equation shown in FIG. 4D:

${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{U\; 1\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}{{P\left( {{\Delta\; f}} \right)}.}}$

Any suitable technique may be used to measure P(|Δf|) representing the frequency response of the actuator P(s) 46 at the frequency |Δf|, and C(|Δf|) representing the frequency response of the servo compensator C(z) 34 at the frequency |Δf|. In one embodiment, the above described anti-aliasing multi-rate (Nx) bode algorithm may be used to measure the frequency response P(|Δf|). However, any suitable algorithm may be employed, including any convention technique for measuring a frequency response of the actuator P(s) 46 at the frequency |Δf|. In one embodiment, the term (1+P(|Δf|)C(|Δf|)) in the above equations may be estimated by adding a sinusoid at the frequency |Δf| to the first control signal u1(k) and evaluating the resulting PES.

Any suitable non-zero value may be selected for the frequency Δf which may be a negative or positive value. In one embodiment, the frequency Δf may be varied from a negative value through zero to a positive value in order to measure the frequency response of the actuator P(s) 46 over a range of frequencies near the servo sample frequency f_(s). In one embodiment, a frequency response of the actuator P(s) 46 may be determined using a conventional algorithm or using the above described anti-aliasing multi-rate (Nx) bode algorithm for frequencies excluding a band near the servo sample frequency f_(s), and then the frequency response may be determined for the missing band using the above described algorithm.

In the embodiments described above, the servo control system such as shown in FIG. 2B processes the discrete time values indexed by k (e.g., PES(k)) at the servo sample frequency f_(s). In other embodiments, the servo control system may employ up-sampling such that the discrete time values indexed by k are processed at a multiple of the servo sample frequency (Nf_(s)). In this embodiment, the first discrete-time sinusoid 40 is generated at the up-sampled frequency and added to the first control signal u1(k) also generated at the up-sampled frequency.

Any suitable control circuitry may be employed to implement the flow diagrams in the above embodiments, such as any suitable integrated circuit or circuits. For example, the control circuitry may be implemented within a read channel integrated circuit, or in a component separate from the read channel, such as a disk controller, or certain operations described above may be performed by a read channel and others by a disk controller. In one embodiment, the read channel and disk controller are implemented as separate integrated circuits, and in an alternative embodiment they are fabricated into a single integrated circuit or system on a chip (SOC). In addition, the control circuitry may include a suitable preamp circuit implemented as a separate integrated circuit, integrated into the read channel or disk controller circuit, or integrated into a SOC.

In one embodiment, the control circuitry comprises a microprocessor executing instructions, the instructions being operable to cause the microprocessor to perform the flow diagrams described herein. The instructions may be stored in any computer-readable medium. In one embodiment, they may be stored on a non-volatile semiconductor memory external to the microprocessor, or integrated with the microprocessor in a SOC. In another embodiment, the instructions are stored on the disk and read into a volatile semiconductor memory when the disk drive is powered on. In yet another embodiment, the control circuitry comprises suitable logic circuitry, such as state machine circuitry.

The various features and processes described above may be used independently of one another, or may be combined in various ways. All possible combinations and subcombinations are intended to fall within the scope of this disclosure. In addition, certain method, event or process blocks may be omitted in some implementations. The methods and processes described herein are also not limited to any particular sequence, and the blocks or states relating thereto can be performed in other sequences that are appropriate. For example, described tasks or events may be performed in an order other than that specifically disclosed, or multiple may be combined in a single block or state. The example tasks or events may be performed in serial, in parallel, or in some other manner. Tasks or events may be added to or removed from the disclosed example embodiments. The example systems and components described herein may be configured differently than described. For example, elements may be added to, removed from, or rearranged compared to the disclosed example embodiments.

While certain example embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the inventions disclosed herein. Thus, nothing in the foregoing description is intended to imply that any particular feature, characteristic, step, module, or block is necessary or indispensable. Indeed, the novel methods and systems described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of the methods and systems described herein may be made without departing from the spirit of the embodiments disclosed herein. 

What is claimed is:
 1. A disk drive comprising: a head; a disk comprising a plurality of servo tracks, wherein each servo track comprises a plurality of servo sectors; an actuator configured to actuate the head over the disk, wherein the actuator comprises a frequency response; control circuitry comprising a servo control system configured to control the actuator, the control circuitry configured to: sample the servo sectors at a servo sample frequency f_(s) to generate a position error signal PES(k); filter the PES(k) with a compensator to generate a first control signal u1(k); add a first discrete-time sinusoid comprising a sinusoid frequency of f_(s)+Δf to the first control signal u1(k) to generate a second control signal u2(k); apply the second control signal u2(k) to the actuator; determine the frequency response of the actuator at the frequency |Δf|; and determine the frequency response of the actuator at the frequency f_(s)+Δf based at least in part on the frequency response of the actuator at the frequency |Δf| and a measured signal of the servo control system when applying the second control signal u2(k) to the actuator.
 2. The disk drive as recited in claim 1, wherein the frequency Δf is one of a positive value and a negative value.
 3. The disk drive as recited in claim 1, wherein the actuator comprises a voice coil motor.
 4. The disk drive as recited in claim 1, wherein the actuator comprises a microactuator.
 5. The disk drive as recited in claim 1, wherein the control circuitry is further configured to determine the frequency response of the actuator at the frequency f_(s)+Δf based on a frequency domain representation of the PES(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator.
 6. The disk drive as recited in claim 5, wherein the control circuitry is further configured to determine the frequency response of the actuator at the frequency f_(s)+Δf based on a second discrete-time sinusoid comprising a sinusoid frequency of |Δf|.
 7. The disk drive as recited in claim 6, wherein the control circuitry is further configured to determine the frequency response of the actuator at the frequency f_(s)+Δf based on: ${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{{PES}\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}\left( {1 + {{P\left( {{\Delta\; f}} \right)}{C\left( {{\Delta\; f}} \right)}}} \right)}$ where: P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf; PES(|Δf|) represents the frequency domain representation of the PES(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator; S₂(|Δf|) represents a frequency domain representation of the second sinusoid; P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|; and C(|Δf|) represents a frequency response of the servo compensator at the frequency |Δf|.
 8. The disk drive as recited in claim 1, wherein the control circuitry is further configured to determine the frequency response of the actuator at the frequency f_(s)+Δf based on a frequency domain representation of the first control signal u1(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator.
 9. The disk drive as recited in claim 8, wherein the control circuitry is further configured to determine the frequency response of the actuator at the frequency f_(s)+Δf based on a second discrete-time sinusoid comprising a sinusoid frequency of |Δf|.
 10. The disk drive as recited in claim 9, wherein the control circuitry is further configured to determine the frequency response of the actuator at the frequency f_(s)+Δf based on: ${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{U\; 1\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}\left\lbrack \frac{1 + {{P\left( {{\Delta\; f}} \right)}{C\left( {{\Delta\; f}} \right)}}}{C\left( {{\Delta\; f}} \right)} \right\rbrack}$ where: P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf; U1(|Δf|) represents the frequency domain representation of the first control signal u1(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator; S₂(|Δf|) represents a frequency domain representation of the second sinusoid; P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|; and C(|Δf|) represents a frequency response of the servo compensator at the frequency |Δf|.
 11. The disk drive as recited in claim 9, wherein the control circuitry is further configured to determine the frequency response of the actuator at the frequency f_(s)+Δf based on: ${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{U\; 1\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}{P\left( {{\Delta\; f}} \right)}}$ where: P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf; U1(|Δf|) represents the frequency domain representation of the first control signal u1(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator; S₂(|Δf|) represents a frequency domain representation of the second sinusoid; and P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|.
 12. A method of operating a disk drive comprising a servo control system for controlling an actuator for actuating a head over a disk, wherein the actuator comprises a frequency response, the method comprising: sampling servo sectors on a disk at a servo sample frequency f_(s) to generate a position error signal PES(k); filtering the PES(k) with a compensator to generate a first control signal u1(k); adding a first discrete-time sinusoid comprising a sinusoid frequency of f_(s)+Δf to the first control signal u1(k) to generate a second control signal u2(k); applying the second control signal u2(k) to the actuator; determining the frequency response of the actuator at the frequency |Δf|; and determining the frequency response of the actuator at the frequency f_(s)+Δf based at least in part on the frequency response of the actuator at the frequency |Δf| and a measured signal of the servo control system when applying the second control signal u2(k) to the actuator.
 13. The method as recited in claim 12, wherein the frequency Δf is one of a positive value and a negative value.
 14. The method as recited in claim 12, wherein the actuator comprises a voice coil motor.
 15. The method as recited in claim 12, wherein the actuator comprises a microactuator.
 16. The method as recited in claim 12, further comprising determining the frequency response of the actuator at the frequency f_(s)+Δf based on a frequency domain representation of the PES(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator.
 17. The method as recited in claim 16, further comprising determining the frequency response of the actuator at the frequency f_(s)+Δf based on a second discrete-time sinusoid comprising a sinusoid frequency of |Δf|.
 18. The method as recited in claim 17, further comprising determining the frequency response of the actuator at the frequency f_(s)+Δf based on: ${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{{PES}\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}\left( {1 + {{P\left( {{\Delta\; f}} \right)}{C\left( {{\Delta\; f}} \right)}}} \right)}$ where: P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf; PES(|Δf|) represents the frequency domain representation of the PES(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator; S₂(|Δf|) represents a frequency domain representation of the second sinusoid; P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|; and C(|Δf|) represents a frequency response of the servo compensator at the frequency |Δf|.
 19. The method as recited in claim 12, further comprising determining the frequency response of the actuator at the frequency f_(s)+Δf based on a frequency domain representation of the first control signal u1(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator.
 20. The method as recited in claim 19, further comprising determining the frequency response of the actuator at the frequency f_(s)+Δf based on a second discrete-time sinusoid comprising a sinusoid frequency of |Δf|.
 21. The method as recited in claim 20, further comprising determining the frequency response of the actuator at the frequency f_(s)+Δf based on: ${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{U\; 1\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}\left\lbrack \frac{1 + {{P\left( {{\Delta\; f}} \right)}{C\left( {{\Delta\; f}} \right)}}}{C\left( {{\Delta\; f}} \right)} \right\rbrack}$ where: P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf; U1(|Δf|) represents the frequency domain representation of the first control signal u1(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator; S₂(|Δf|) represents a frequency domain representation of the second sinusoid; P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|; and C(|Δf|) represents a frequency response of the servo compensator at the frequency |Δf|.
 22. The method as recited in claim 20, further comprising determining the frequency response of the actuator at the frequency f_(s)+Δf based on: ${P\left( {f_{s} + {\Delta\; f}} \right)} = {\frac{U\; 1\left( {{\Delta\; f}} \right)}{S_{2}\left( {{\Delta\; f}} \right)}{P\left( {{\Delta\; f}} \right)}}$ where: P(f_(s)+Δf) represents the frequency response of the actuator at the frequency f_(s)+Δf; U1(|Δf|) represents the frequency domain representation of the first control signal u1(k) at the frequency |Δf| when applying the second control signal u2(k) to the actuator; S₂(|Δf|) represents a frequency domain representation of the second sinusoid; and P(|Δf|) represents the frequency response of the actuator at the frequency |Δf|. 